Greater than 1L.kg-1 indicates a drug is highly protein bound or lipophilicAgents which cross the blood brain barrier typically have a VD of 1-2L.kg-1.

Subtypes of the volume of distribution are used to describe drug distribution at different times or with different models

These include:

V1Volume of central compartment.

VDssVolume of distribution at steady state.

VDpeVolume of distribution at peak effect.

Which volume to use depends on the pharmacological question

e.g. Intubating dose for opioid should use a volume between V1 (very small) and VDss (very large) - VDpe is ideal as it will allow a target concentration to be selected for the time at which intubation will occur relative to drug administration

Half-life (t1/2)The time it takes for a process to be 50% complete. With respect to drug clearance, it is the time it takes for concentration (typically in plasma) to fall by 50%.

A process is considered to be complete after 4-5 half-livesConcentration will decrease by 50% after each half-life, so after 5 half-lives concentration will be 3.125% of its starting value.

This also applies to wash in - it will take ~4-5 elimination half-lives of a drug for a constant-rate infusion to reach its final concentration

Half-life is mathematically related to many other key pharmacokinetic terms:, where:

is the time constant

is the rate constant for elimination

is the volume of distribution

is the clearance

Various types of half-life are described:

t1/2α describes the rapidity of the distribution phase following drug administration

t1/2β describes the rapidity of the elimination phase occurring after drug distribution equilibriumThis only evaluates clearance from plasma, and so is a composite of both excretion from the body (e.g. renal and hepatic clearance) and ongoing distribution to peripheral tissues.

The elimination half-life is generally not useful to predict drug offset, as this is affected by many factorsHowever, it does set an upper limit on how long it will take plasma concentration to fall by 50%.

Time-constant ()The time taken for a process to complete if it continued at its initial rate of change. Time constants are related to half-life, but are better suited when modelling change in exponential processes.

A time constant is the inverse of the rate constant for elimination, i.e.

Illustration of the relationship between half-life and time constant:

ClearanceThe clearance is volume of plasma completely cleared of a drug per unit time.

In a one compartment model, this can be expressed as: in ml.min-1.

As the time constant is the inverse of k, clearance can also be expressed as:

Since and are constants, clearance is also a constant

Total clearance is a sum of the clearance of each individual clearance organ

Rate of eliminationAmount of drug removed by the body per unit time.

Rate of elimination is the product of the clearance and the current concentration: , in mg.min-1

This is not the rate constant for elimination

Compartmental Modelling

The simplest model imagines the body a single, well-stirred compartment.

In a one compartment model, the concentration of a drug () at time is given by the equation:

Where:

is the concentration at time 0As drug can only be eliminated from the compartment, this is also the peak concentration.

k is the rate constant for eliminationThis is the fraction of the Vd from which the drug is removed per unit time. The rate constant determines the slope of the curve.

A high rate constant for elimination results in a steep curve and therefore a short time constant

Steady state

At steady state, input is equal to output. Therefore concentration at steady state is:

Proportional to the concentration of the infusion and infusion rate

Inversely proportional to the clearance:

Concentration of drug can therefore be determined by the amount infused and the clearance

Note steady state requires peripheral compartments to be saturated, and so will only occur after an infusion of many hours

Multiple Compartment Models

Models with multiple compartments have a better fit with experimental data than single compartment models

Three-compartment models are typically used, as additional compartments typically offer no extra fidelity but are mathematically more complex

A three-compartment model can be conceptualised as a plasma (or central) compartment, a well-perfused compartment, and a poorly-perfused compartmentThis doesn't mean that they should be thought of in this way - they are a mathematical technique used to calculate plasma concentration at a given time.

Plasma concentration in multicompartment models is:

Predicted through the net effect of several negative exponential equations x
This is covered under two-compartment models below.

Dependent on the effects of:

DistributionDistribution describes the movement of drug from the central compartment (V1) to the peripheral compartment(s).

Rapid fall in plasma concentration of a drug after administration is generally due to distributionDistribution is an important method for drug offset in short-acting drugs.

RedistributionRedistribution refers to the movement of drug from the peripheral compartment(s) back into plasma.

Drugs which have a large VD in a peripheral compartment tend to distribute quickly along this concentration gradient, and redistribute slowly back into plasma

Drugs which tend to distribute slowly tend to redistribute quickly once administration has ceased

ExcretionExcretion is the removal of drug from the body.

Clearance in Two-Compartment Models

Removal of drug in two-compartment models is via:

Distribution from the central to the peripheral compartment

Elimination from the central compartment

This produces a bi-exponential fall in plasma concentrationConsists of two phases:

Phase αDistribution phase: A rapid decline in plasma concentration due to distribution to peripheral tissues.

is the y-intercept of the distribution exponentUsed to calculate distribution half-life.

is the y-intercept of the elimination exponentUsed to calculate elimination half-life.

is the rate constant for distributionThe value of is dependent on the ratio of rate constants for distribution and redistribution (i.e. ).

If distribution greatly exceeds redistribution, the gradient of will be very steep and plasma concentration will fall rapidly after administration

is the rate constant for elimination

Note that the distribution and elimination curves appear straight because the y-axis is log-transformed

If plasma concentration was plotted on the y-axis, then each of these curves would be a negative exponential (wash-out curve)

Effect Site

Pharmacokinetic models typically display the plasma concentration.

Clinically however, we are interested in drug concentrations at the site of action (e.g. the brain)

Concentration at the effect site (also known as biophase) is given by Ce

This cannot be measured, and so is a calculated value

Effect site concentration be different from plasma concentration (Cp) prior to reaching steady stateThe delay between plasma and effect site concentrations is an example of hysteresis.

The effect site can be modelled as an additional compartment in three-compartment modelsThe effect site is modelled as a compartment of negligible volume contained within V1, but does have rate constants

Effect site volume changes as V1 changes

The ke1 is the rate constant for drug diffusion from plasma into the effect site

The ke0 is the rate constant for elimination of drug from the effect siteThis is a theoretical elimination pathway - drug is not usually metabolised at the effect site.

The t1/2ke0 describes the effect-site equilibration timeIt describes how rapidly the effect site reaches equilibrium with plasma.

A large ke0 (rapid drug flow) gives a short t1/2ke0

After one t1/2ke0, 50% of the final effect site concentration will be reached provided plasma concentration remains constant

A shorter t1/2ke0 indicates that that the effect site concentration will reach equilibrium with plasma more rapidly, and therefore a more rapid clinical effect following administration is seen

Note that:

The t1/2ke0 is not the time to peak effect

Neither is ke0

For an infusion run at constant plasma concentration the peak effect will be seen at 3-5x the t1/2ke0

The time to peak effect is a function of both plasma kinetics and the t1/2ke0

e.g. adenosine has such a short elimination t1/2 the effect site concentration will reach its peak rapidly regardless of the ke0

Non-Compartmental Models

Compartment models are not appropriate for describing the behaviours of all drugs. Non-compartmental models are used when drug:

Clearance is organ-independent

Elimination does not occur solely from the central compartment

These models use AUC, which is calculated by measuring the plasma concentration of a drug at different time intervals, and plotting the area under the curve (AUC). This can be used to:

Determine clearance

Determine BioavailabilityDifference between the AUC of the same dose of drug administered IV and via another route.

Footnotes

The formula for half-life can be derived from the equation for a wash-in exponential as follows: